Cartesian perfectly matched layers for the Helmholtz equation with variable propagation speed: well-posedness and exponential convergence
Abstract
We analyse the Cartesian perfectly matched layer (PML) approximation of the two-dimensional Helmholtz equation −c2Δu−ω2 u=f, in which the propagation speed c is variable inside the region of interest and equal to a constant c∞ outside a disc BR. Existing analyses of the Cartesian PML assume a homogeneous, or piecewise homogeneous, background. We assume of c only a two-sided bound, so that discontinuous speeds -- layered media and compact inclusions -- are covered. This is possible because the speed enters the PML sesquilinear form only through its zeroth order coefficient, the principal part being built from the stretching profiles alone. Well-posedness of the physical problem at this regularity follows from a Lippmann-Schwinger equation, uniqueness from Rellich's lemma and unique continuation. A Morawetz multiplier yields a sharp non-trapping condition on c, namely ∇ c(x)·(x−x0)<c(x) for some x0, which reduces in the radial case to the Herglotz condition d/dρ(ρ/c(ρ))>0; its failure produces a trapped bicharacteristic. The variable speed is invisible to the essential spectrum: after factoring out c2, the two operators differ by a compactly supported multiplication, so the spectral analysis of the homogeneous Cartesian layer applies verbatim. This yields well-posedness of the truncated problem for every ω>0, with a stability constant independent of the layer width, and convergence to the physical solution in H1 at a rate exponential in ωγMδM, where γM and δM are the absorption strength and the layer thickness. Numerical experiments with a low-velocity inclusion confirm the exponential decay in both δ and ω.
Related
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BibTeX
@misc{ortizsaavedra2026cartesian,
title = {{Cartesian perfectly matched layers for the Helmholtz equation with variable propagation speed: well-posedness and exponential convergence}},
author = {César Ortiz-Saavedra and Mauricio A. Londoño-Arboleda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20992},
eprint = {2609.20992},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20992},
}
RIS
TY - UNPB
TI - Cartesian perfectly matched layers for the Helmholtz equation with variable propagation speed: well-posedness and exponential convergence
AU - César Ortiz-Saavedra
AU - Mauricio A. Londoño-Arboleda
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20992
UR - https://arxiv.org/abs/2609.20992
AB - We analyse the Cartesian perfectly matched layer (PML) approximation of the two-dimensional Helmholtz equation −c^2Δu−ω^2 u=f, in which the propagation speed c is variable inside the region of interest and equal to a constant c_∞ outside a disc B_R. Existing analyses of the Cartesian PML assume a homogeneous, or piecewise homogeneous, background. We assume of c only a two-sided bound, so that discontinuous speeds -- layered media and compact inclusions -- are covered. This is possible because the speed enters the PML sesquilinear form only through its zeroth order coefficient, the principal part being built from the stretching profiles alone. Well-posedness of the physical problem at this regularity follows from a Lippmann-Schwinger equation, uniqueness from Rellich's lemma and unique continuation. A Morawetz multiplier yields a sharp non-trapping condition on c, namely ∇ c(x)·(x−x_0)<c(x) for some x_0, which reduces in the radial case to the Herglotz condition d/dρ(ρ/c(ρ))>0; its failure produces a trapped bicharacteristic. The variable speed is invisible to the essential spectrum: after factoring out c^2, the two operators differ by a compactly supported multiplication, so the spectral analysis of the homogeneous Cartesian layer applies verbatim. This yields well-posedness of the truncated problem for every ω>0, with a stability constant independent of the layer width, and convergence to the physical solution in H^1 at a rate exponential in ωγ_Mδ_M, where γ_M and δ_M are the absorption strength and the layer thickness. Numerical experiments with a low-velocity inclusion confirm the exponential decay in both δ and ω.
ER -